*Nicholas Manton and Nicholas Mee*

- Published in print:
- 2017
- Published Online:
- July 2017
- ISBN:
- 9780198795933
- eISBN:
- 9780191837111
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198795933.003.0006
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter develops the mathematical technology required to understand general relativity by taking the reader from the traditional flat space geometry of Euclid to the geometry of Riemann that ...
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This chapter develops the mathematical technology required to understand general relativity by taking the reader from the traditional flat space geometry of Euclid to the geometry of Riemann that describes general curved spaces of arbitrary dimension. The chapter begins with a comparison of Euclidean geometry and spherical geometry. The concept of the geodesic is introduced. The discovery of hyperbolic geometry is discussed. Gaussian curvature is defined. Tensors are introduced. The metric tensor is defined and simple examples are given. This leads to the use of covariant derivatives, expressed in terms of Christoffel symbols, the Riemann curvature tensor and all machinery of Riemannian geometry, with each step illustrated by simple examples. The geodesic equation and the equation of geodesic deviation are derived. The final section considers some applications of curved geometry: configuration space, mirages and fisheye lenses.Less

This chapter develops the mathematical technology required to understand general relativity by taking the reader from the traditional flat space geometry of Euclid to the geometry of Riemann that describes general curved spaces of arbitrary dimension. The chapter begins with a comparison of Euclidean geometry and spherical geometry. The concept of the geodesic is introduced. The discovery of hyperbolic geometry is discussed. Gaussian curvature is defined. Tensors are introduced. The metric tensor is defined and simple examples are given. This leads to the use of covariant derivatives, expressed in terms of Christoffel symbols, the Riemann curvature tensor and all machinery of Riemannian geometry, with each step illustrated by simple examples. The geodesic equation and the equation of geodesic deviation are derived. The final section considers some applications of curved geometry: configuration space, mirages and fisheye lenses.

*Glen Van Brummelen*

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691175997
- eISBN:
- 9781400844807
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175997.003.0002
- Subject:
- Mathematics, History of Mathematics

This chapter introduces the reader to the celestial sphere, or the Earth's surface. By rotating the sphere, the motions of the heavens can be simulated. There are three features of celestial motion ...
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This chapter introduces the reader to the celestial sphere, or the Earth's surface. By rotating the sphere, the motions of the heavens can be simulated. There are three features of celestial motion that came to be associated with Aristotle: all objects move in circles; they travel at constant speeds on those circles; the Earth is at the center of the celestial sphere. The chapter shows how the movements of stars and planets on the sphere's surface can be determined by setting up a system of equatorial coordinates. It also explains how the celestial sphere can be set in motion through the day, and how Hipparchus of Rhodes endeavored to determine the eccentricity of the Sun's orbit. Finally, it discusses spherical geometry, with emphasis on finding bounds on the sides and angles of a spherical triangle.Less

This chapter introduces the reader to the celestial sphere, or the Earth's surface. By rotating the sphere, the motions of the heavens can be simulated. There are three features of celestial motion that came to be associated with Aristotle: all objects move in circles; they travel at constant speeds on those circles; the Earth is at the center of the celestial sphere. The chapter shows how the movements of stars and planets on the sphere's surface can be determined by setting up a system of equatorial coordinates. It also explains how the celestial sphere can be set in motion through the day, and how Hipparchus of Rhodes endeavored to determine the eccentricity of the Sun's orbit. Finally, it discusses spherical geometry, with emphasis on finding bounds on the sides and angles of a spherical triangle.

*Yunping Jiang*

*Araceli Bonifant, Mikhail Lyubich, and Scott Sutherland (eds)*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159294
- eISBN:
- 9781400851317
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159294.003.0012
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter reviews the characterization of geometrically finite rational maps and then outlines a framework for characterizing holomorphic maps. Whereas Thurston's methods are based on estimates of ...
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This chapter reviews the characterization of geometrically finite rational maps and then outlines a framework for characterizing holomorphic maps. Whereas Thurston's methods are based on estimates of hyperbolic distortion in hyperbolic geometry, the framework suggested here is based on controlling conformal distortion in spherical geometry. The new framework enables one to relax two of Thurston's assumptions: first, that the iterated map has finite degree and, second, that its post-critical set is finite. Thus, it makes possible to characterize certain rational maps for which the post-critical set is not finite as well as certain classes of entire and meromorphic coverings for which the iterated map has infinite degree.Less

This chapter reviews the characterization of geometrically finite rational maps and then outlines a framework for characterizing holomorphic maps. Whereas Thurston's methods are based on estimates of hyperbolic distortion in hyperbolic geometry, the framework suggested here is based on controlling conformal distortion in spherical geometry. The new framework enables one to relax two of Thurston's assumptions: first, that the iterated map has finite degree and, second, that its post-critical set is finite. Thus, it makes possible to characterize certain rational maps for which the post-critical set is not finite as well as certain classes of entire and meromorphic coverings for which the iterated map has infinite degree.